Answer:
b. [tex]4x^2+3y^2-24y-144=0[/tex]
Step-by-step explanation:
The given equation is;
[tex]r=\frac{-72}{12+6\sin(\theta)}[/tex]
Cross multiply;
[tex]r(12+6\sin(\theta)=-72[/tex]
Expand;
[tex]12r+6r\sin(\theta)=-72[/tex]
Divide through by 6;
[tex]2r+r\sin(\theta)=-12[/tex]
[tex]2r=-12-r\sin(\theta)[/tex]
Substitute;
[tex]y=r\sin(\theta), r=\sqrt{x^2+y^2}[/tex]
[tex]2\sqrt{x^2+y^2}=-12-y[/tex]
Square both sides;
[tex](2\sqrt{x^2+y^2})^2=(-12-y)^2[/tex]
[tex]4(x^2+y^2)=144+24y+y^2[/tex]
[tex]4x^2+4y^2=144+24y+y^2[/tex]
Simplify;
[tex]4x^2+3y^2-24y-144=0[/tex]
